The magnitude of the impedance in a series RLC circuit is given by which expression?

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Multiple Choice

The magnitude of the impedance in a series RLC circuit is given by which expression?

Explanation:
In a series RLC circuit, impedance is a complex quantity with a real part equal to the resistance and an imaginary part equal to the net reactance. The net reactance is the difference between the inductive and capacitive reactances, XL − XC, because the inductor contributes +jXL and the capacitor contributes −jXC in the impedance. So Z = R + j(XL − XC). The magnitude of impedance is the length of that complex number, which is obtained by the Pythagorean relation: |Z| = sqrt(R^2 + (XL − XC)^2). This is why the minus sign matters here: XC acts to reduce the net reactance, not add to it, when you consider impedance. Other forms would misrepresent the vector nature of impedance. For example, adding XL and XC directly ignores that XC is a negative imaginary component, and taking R plus the reactance magnitude or using a minus under the square root would not yield the correct, nonnegative magnitude.

In a series RLC circuit, impedance is a complex quantity with a real part equal to the resistance and an imaginary part equal to the net reactance. The net reactance is the difference between the inductive and capacitive reactances, XL − XC, because the inductor contributes +jXL and the capacitor contributes −jXC in the impedance. So Z = R + j(XL − XC).

The magnitude of impedance is the length of that complex number, which is obtained by the Pythagorean relation: |Z| = sqrt(R^2 + (XL − XC)^2). This is why the minus sign matters here: XC acts to reduce the net reactance, not add to it, when you consider impedance.

Other forms would misrepresent the vector nature of impedance. For example, adding XL and XC directly ignores that XC is a negative imaginary component, and taking R plus the reactance magnitude or using a minus under the square root would not yield the correct, nonnegative magnitude.

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